OFFSET
1,1
COMMENTS
Conjecture: For any s and t in the set {1,-1}, every positive rational number r can be written as m/n with m and n in the set {k>0: prime(prime(k))+s*prime(k)+t = prime(p) for some prime p}.
In the case s = -1 and t = 1, the conjecture implies that A261136 has infinitely many terms.
REFERENCES
Zhi-Wei Sun, Problems on combinatorial properties of primes, in: M. Kaneko, S. Kanemitsu and J. Liu (eds.), Number Theory: Plowing and Starring through High Wave Forms, Proc. 7th China-Japan Seminar (Fukuoka, Oct. 28 - Nov. 1, 2013), Ser. Number Theory Appl., Vol. 11, World Sci., Singapore, 2015, pp. 169-187.
LINKS
Zhi-Wei Sun, Table of n, a(n) for n = 1..5000
Zhi-Wei Sun, Problems on combinatorial properties of primes, arXiv:1402.6641 [math.NT], 2014.
EXAMPLE
a(3) = 2279 since prime(prime(2279))-prime(2279)+1 = prime(20147)-20147+1 = 226553-20146 = 206407 = prime(18503) with 18503 prime, and prime(prime(2279*3))-prime(2279*3)+1 = prime(68777)-68777+1 = 865757-68776 = 796981 = prime(63737) with 63737 prime.
MATHEMATICA
f[n_]:=Prime[Prime[n]]-Prime[n]+1
PQ[p_]:=PrimeQ[p]&&PrimeQ[PrimePi[p]]
Do[k=0; Label[bb]; k=k+1; If[PQ[f[k]]&&PQ[f[k*n]], Goto[aa], Goto[bb]]; Label[aa]; Print[n, " ", k]; Continue, {n, 1, 60}]
CROSSREFS
KEYWORD
nonn
AUTHOR
Zhi-Wei Sun, Aug 18 2015
STATUS
approved